We now present a sparse model that is computationally tractable in terms of inference and prediction with GPs. The sparsity arises because we consider a sparse dataset \(\bar{D}\)  of size \(m<n\)  with inducing inputs \(\bar{\Theta } = \lbrace \bar{\theta _i},~i=1,\ldots ,m\rbrace  \subset \mathcal {T}\) . These inducing inputs can either be a subset of the training inputs or can be randomly sampled from \(\mathcal {T}\)  [1]}. Let \(u=f(\bar{\theta })\)  be the inducing outputs, which are sampled from the same prior on the true function \(f\) . In this work, we treat \(\bar{\Theta }\)  as hyperparameters and estimate them from maximizing the marginal likelihood.
